ppl ath vol. 78, no. 1, pp. 418{436 motivated by coffee

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SIAM J. APPL. MATH. c© 2018 Society for Industrial and Applied MathematicsVol. 78, No. 1, pp. 418–436

ASYMPTOTIC ANALYSIS OF A MULTIPHASE DRYING MODELMOTIVATED BY COFFEE BEAN ROASTING∗

NABIL T. FADAI† , COLIN P. PLEASE† , AND ROBERT A. VAN GORDER†

Abstract. Recent modeling of coffee bean roasting suggests that in the early stages of roasting,within each coffee bean, there are two emergent regions: a dried outer region and a saturatedinterior region. The two regions are separated by a transition layer (or, drying front). In thispaper, we consider the asymptotic analysis of a recent multiphase model in order to gain a betterunderstanding of its salient features. The model consists of a PDE system governing the thermal,moisture, and gas pressure profiles throughout the interior of the bean. By obtaining asymptoticexpansions for these quantities in relevant limits of the physical parameters, we are able to determinethe qualitative behavior of the outer and interior regions, as well as the dynamics of the drying front.Although a number of simplifications and scalings are used, we take care not to discard aspectsof the model which are fundamental to the roasting process. Indeed, we find that for all of theasymptotic limits considered, our approximate solutions faithfully reproduce the qualitative featuresevident from numerical simulations of the full model. From these asymptotic results, we have abetter qualitative understanding of the drying front (which is hard to resolve precisely in numericalsimulations) and, hence, of the various mechanisms at play including heating, evaporation, andpressure changes. This qualitative understanding of solutions to the multiphase model is essentialwhen creating more involved models that incorporate chemical reactions and solid mechanics effects.

Key words. multiphase model, coffee bean roasting, Stefan problem, asymptotic analysis,drying front

AMS subject classifications. 80A22, 80M35, 74N20, 82C26

DOI. 10.1137/16M1095500

1. Introduction. As one of the most valuable commodities in the world [1],the coffee industry relies on fundamental research to improve the techniques andprocesses relating to its products, especially with the roasting process of coffee beans.Most of the literature concerning the roasting of coffee beans present experimentaldata (see, e.g., [2, 3, 4]) and use regression analysis and simple empirical models tointerpret the results. Recently, the literature has included a more in-depth discussionconcerning the mathematical modeling of the roasting of coffee beans (see, e.g., [5, 6]).While other aspects of coffee processing have been examined from a mathematicalperspective (e.g., [7]), mathematical models describing the roasting of coffee beanshave, with the exception of a few studies, been largely unexplored.

In [5], a system of partial differential equations (PDEs) that model the transportof moisture and heat throughout a coffee bean were derived and studied. This modeluses the concept of “mass diffusivity” to describe the transport of moisture in thecoffee bean that was originally derived in [8], which applies to lower-temperatureevaporation. The ideas in [5] served as excellent motivation for the authors in [6]to derive a mathematical model from first principles using conservation equations.Multiphase modeling has previously been applied in a variety of food heating problems

∗Received by the editors September 26, 2016; accepted for publication (in revised form) October30, 2017; published electronically February 6, 2018.

http://www.siam.org/journals/siap/78-1/M109550.htmlFunding: This publication is based on work supported by the EPSRC center for Doctoral

Training in Industrially Focused Mathematical Modelling (EP/L015803/1).†Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observa-

tory Quarter, Oxford, OX2 6GG, UK ([email protected], [email protected],[email protected]).

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http://www.siam.org/journals/siap/78-1/M109550.html

mailto:[email protected]




ASYMPTOTIC ANALYSIS OF A MULTIPHASE DRYING MODEL 419

[9, 10, 11, 12, 13, 14] (of particular relevance was the bread baking model of Zhangand Datta [15]) and is a natural framework to model the coffee bean roasting process.In [6], the concepts of multiphase flow and water evaporation were included, and theresulting model (referred to as Model 2 in [6]) also incorporated the production ofcarbon dioxide gas, latent heat due to evaporation, and changes in porosity. Somesimplifications were made to this full model (in particular, neglecting carbon dioxideproduction) in order to allow some preliminary understanding of the model behavior.By examining numerical solutions of this model, a “drying front” that propagatesthrough to the center of the bean was identified.

Moisture transport in roasting coffee beans is a crucial aspect of the industrialprocess. Many chemical reactions that occur produce aromas and flavor compoundsthat determine the quality of the final product. Some of these chemical reactions arebelieved to be affected by the presence (or lack) of water [3]. Therefore, the spatialand temporal distribution of the moisture content is strongly linked to the quality ofthe final product.

Mathematical models describing drying have been explored previously (see, e.g.,[16, 17, 18]), which relate the drying of wood, bricks, and other materials. However,due to the impermeable cellulose structure within a coffee bean, the water vapor cre-ated in the biological cells cannot be easily released into the roasting environment. Inconsequence, the ratio in time scales between evaporation dynamics and vapor trans-port is very large, which motivates us to explore the leading-order dynamics of coffeebean roasting using asymptotic analysis. Hence, the model presented here should beappropriate to any drying problem where these physical phenomena are relevant.

In [6], numerical results suggest that there are three main regions within a coffeebean as it is roasted. The first main region (which we refer to as Region i) is where thevapor pressure of water aligns with the steam table pressure. The second main region(which we refer to as Region ii) is when the moisture content of the bean is negligible.Between these two regions, we expect a thin transition layer, or drying front, inwhich moisture is rapidly evaporated. Issues surrounding numerical resolution makeit computationally expensive to resolve the dynamics near the drying front. In light ofthese observations, we are motivated to extend the numerical results shown in [6] viaasymptotic methods, in order to understand the qualitative features of the multiphasemodel: in particular, the interplay between the narrow transition layer and the twolarger regions.

In this paper, we begin our discussion of the asymptotics of the full multiphasemodel in section 2. Motivated by the numerical results in [6], we determine an ap-proximate form of the drying front. We then obtain the leading-order asymptotics inRegions i and ii, as well as within the drying front. Despite several simplifications,we are able to obtain reasonable agreement between the asymptotic approximationsand the numerical solutions in [6], and are confident that the asymptotics capture thequalitative dynamics of the problem. In order to obtain additional explicit results, andmotivated by the observation that the entire coffee bean is almost always very close tothe externally imposed roasting temperature, we fix the temperature at this roastingtemperature in section 3. Under this assumption, the vapor pressure and moisturecontent are also constant in Region i, while the leading order dynamics within Regionii reduce to a Stefan problem [19, 20]. By considering the large Stefan number limit,we determine a leading-order expression for the drying front for various geometries.As we will only focus on symmetric geometries with one spatial variable (e.g., planarand spherical geometries), we can obtain explicit expressions for the drying front. Wefocus on the planar and spherical geometries, as it is reasonable to represent a coffee

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420 N. T. FADAI, C. P. PLEASE, AND R. A. VAN GORDER

bean either as a slab of porous material “curled up” into the shape of a bean or as asphere. Finally, in section 4, we provide a summary and discussion of the results.

2. Asymptotics of the multiphase model with variable temperature.The full multiphase model that we will analyze consists of three PDEs that describeconservation of mass in water and vapor, as well as conservation of energy. Theseequations describe the behavior of three variables: the water saturation S (i.e., thevolume fraction of water divided by the total volume of water and gas), the partialpressure of water vapor P , and the temperature T . The only transport mechanismconsidered for water is via evaporation, whereas in the gas phase, water vapor istransported either via evaporation or via Darcy flow. Finally, we assume that heatis transported via conduction in all three phases within the bean, but via convectionat the surface of the bean. Mathematically speaking, this multiphase model canbe stated for nondimensional variables in symmetric geometries (i.e., using a singlespatial variable r) as

∂S

∂t= − 1

ε2Iv,(1)

∂

∂t

[(1 + T )P (1− σS)

1 + T T

]= −1

δ

∂S

∂t+∇ ·

[(1 + T )P∇P

1 + T T

],(2)

∂T

∂t+A1

∂

∂t[S(1 + T T )] = A2

∂S

∂t+A3∇ · [(1 +A4S)∇T ] ,(3)

where the ∇ operator should be interpreted appropriately for the different geometriesas derivatives of r. Additionally, there are the symmetry conditions at the center ofthe bean,

(4) ∇T · n|r=0 = 0, ∇P · n|r=0 = 0,

the boundary conditions at the surface of the bean,

(5) ∇T · n|r=1 = ν

(1− σS1− σ

)(1 +A4

1 +A4S

)(1− T ) ,

(6) P |r=1 =

{PST (T ), T < Ta,

Pa, T ≥ Ta,

and the initial conditions

(7) S(r, 0) = 1, T (r, 0) = 0, P (r, 0) = PST (0).

Here, the evaporation rate Iv and steam table pressure PST (T ) are given by

(8) Iv = S(1− σS)(PST − P )

√1 + T

1 + T Tand PST (T ) = exp

(β(T − 1)

1 + T T

).

A complete derivation of this model from the “simplified” multiphase model presentedin [6] can be found in Appendix A. A key feature of this model is ε, which canbe interpreted as a ratio in time scales between Darcy-driven vapor transport andevaporation. One can interpret δ as a density ratio of water vapor to water, andσ represents the initial water-to-void volume ratio. The boundary condition (6) is

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Fig. 1. A summary of where the different regions are as the bean dries. Region i is wherethe vapor pressure is in equilibrium, Region ii is the dry region, and the dashed lines indicate thenarrow transition layer between the regions, which begins at time t∗, defined in (9).

slightly modified from that in [6] and is described in Appendix A. Here, Pa is theambient vapor pressure in the roasting chamber.

We will also make the assumption that the step in the boundary condition (6) forP only occurs at one critical time, namely, t∗. We define t∗ as the first time when theevaporation temperature Ta is achieved at the surface of the bean, i.e., as the solutionto the equation

(9) T (1, t∗) = Ta := P−1ST (Pa) .

This critical time will be used not only to signal which part of the step in (6) isrelevant, but also to signal where the asymptotic behavior changes.

We can divide the solution to the model into three regions in order to understandthe approximate dynamics that occur in the coffee bean. Using parameter valuesshown in [6], a typical value of ε ≈ 1.54× 10−4 suggests that we should consider thelimit as ε→ 0+. We note that δ = O(1) is the distinguished limit of this system andconcentrate on considering this case, despite the analysis also being valid for small δ.We will consider δ being small when the equations in Regions i and ii require furthersimplifications to extract closed form solutions. In section 2.2, we will determine howsmall δ is allowed to be before a different analysis is required. In the ε→ 0+ limit, wecan see from (1) that if time and space remain unscaled, Iv = 0 will be the leading-order equation, and from (8), this can occur in one of three ways. First, Iv = 0 ifthe vapor pressure is in equilibrium with its steam table pressure, i.e., P = PST . Asthe initial data are consistent with this equilibrium, we will observe this first (whichwill be referred to as Region i). Second, Iv = 0 can be achieved by setting S = 0.This corresponds to where there is no more water to evaporate and will be denotedas Region ii. A final case where Iv = 0 is when S = σ−1; however, this correspondsto when the coffee bean is completely saturated with water, which we will discard asan extraneous case.

We will also consider a narrow drying front that connects Regions i and ii. Thisdrying front, which is centered about r = R(t), propagates from the surface of thebean towards the center of the bean and is where the moisture content S quickly goesfrom 1 to 0. In this drying front around R(t), we find that the temperature is spatiallyuniform, but will vary as time progresses. The temperature profile within the dryingfront is denoted as T ∗(t). A schematic diagram of these three regions is shown inFigure 1, including the time t∗ at which evaporation first occurs at the surface of thebean.

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2.1. Asymptotics of Region i. In Region i, we have, from (1), Iv = 0 toleading order in the limit as ε → 0+, implying that that P = PST (T ). Consider theasymptotic series valid as ε→ 0+,

S = S0(r, t) + εS1(r, t) +O(ε2),(10)

T = T0(r, t) + εT1(r, t) +O(ε2),

Pv = PST (r, t) + εP1(r, t) +O(ε2).

Substituting (11) into (2) and (3) gives, to lowest order,

∂S0

∂t

(1

δ− σPST (T0)Λ(T0)

)+ (1− σS0)

∂

∂t[PST (T0)Λ(T0)] = ∇ · [PST (T0)Λ(T0)∇PST (T0)] ,(11)

∂S0

∂t[A1(1 + T T0)−A2] +

∂T0

∂t[1 +A1T S0] = A3∇ · [(1 +A4S0)∇T0] ,(12)

where Λ(T0) = 11+T T0

. As we cannot solve this system analytically, we now supposethat δ � 1 and write an asymptotic series in powers of δ for S0 and T0 valid in thelimit δ → 0+ as

(13) S0 = S0(r, t) + δS1(r, t) +O(δ2), T0 = T0(r, t) + δT1(r, t) +O(δ2).

Substituting (13) into (11) gives us, to leading order, that ∂S0

∂t = 0. Therefore, the

moisture content of the bean stays at its initial value, i.e., S0 = 1. To lowest order,(12) then gives us

(14)∂T0

∂t= K∇2T0, where K =

A3(1 +A4)

1 +A1T.

Equation (14) must be solved subject to the boundary condition (6), which can bestated as

∇ · T0

∣∣∣r=1

= ν(

1− T0

∣∣r=1

), t < t∗,(15)

T0

∣∣r=R(t)

= T ∗(t), t ≥ t∗.(16)

Additionally, we will impose the symmetry condition ∇T0 · n = 0 at r = 0, as well asthe initial data T0(r, 0) = 0. We are able to solve the PDE for t < t∗; in particular,we can determine a leading-order approximation for t∗. By solving (14) in sphericalcoordinates, we obtain that

(17) T0(r, t) = 1−∞∑n=1

cnr

sin(µnr) exp(−µ2nKt),

where the eigenvalues µn satisfy the transcendental equation µn cot(µn) = 1− ν andthe constants cn have the form

(18) cn =

2ν cosµn

µn(sin2 µn−ν), ν 6= 1,

8(−1)n

π2(1+2n)2 , ν = 1.

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While the case ν = 1 is reasonably pathological, it is included in this analysis forcompleteness. Typical parameter values seen in [6] give ν ≈ 0.585, so we will onlyconsider the case where ν 6= 1 from here on.

To determine t∗ in spherical coordinates, denoted as t∗Sph, we impose, from (9),

that T0(1, t∗Sph) = Ta. When ν 6= 1, this is equivalent to writing

(19)

∞∑n=1

(cos2 µn

sin2 µn − ν

)exp(−µ2

nKt∗Sph) =(1− Ta) (1− ν)

2ν.

Using the parameter values in [6] (K ≈ 2.25, ν ≈ 0.585, Ta ≈ 0.519), (19) has thesolution t∗Sph ≈ 0.173, or about 45.9 seconds in dimensional units.

Similarly, we can determine t∗ in Cartesian coordinates, denoted as t∗Cart, bynoting that the solution of (14) in Cartesian coordinates with a Neumann boundarycondition at r = 0 is

(20) T0(r, t) = 1−∞∑n=1

dn cos(λnr) exp(−λ2nKt),

where

(21) λn tanλn = ν and dn =2ν sinλn

λn(ν + sin2 λn).

This allows us to determine t∗Cart via the transcendental equation

(22)

∞∑n=1

sin2 λn

ν + sin2 λnexp(−λ2

nKt∗Cart) =1− Ta

2,

which, using parameter values stated above, gives t∗Cart ≈ 0.494, or about 131 sec-onds in dimensional units. Hence, we have determined when the step occurs in theboundary condition (6). When the boundary condition changes, we find that thetransition layer forms at the surface. We shall not consider the very small time whilethe transition layer is close to the surface, as it does not give any useful insight, butproceed to when it is in the interior of the bean.

2.2. Asymptotics of the transition layer. In order to understand how Svaries from 1 to 0, we must examine the transition layer in the limit ε → 0+. Wefirst discuss the distinguished limit δ = O(1). We will find that both P and T arespatially uniform at lowest order with small variations of equal size (say, O(εa), a > 0)and hence, variations of the evaporation rate are also O(εa). The transition layer isnarrow (say, O(εb), b > 0) around a moving front at r = R(t) and we find, notingthat S = O(1), that a balance between the resulting advective term in (1) and theevaporation rate requires −b = −2+a. Similarly, having a balance of vapor productiondue to evaporation with transport by Darcy flow in (2) requires that −b = −2b + a.Hence, in the distinguished limit δ = O(1), we take a = b = 1. We can also considersmall δ; when δ = O(εα) and α > 0, a dominant balance requires that −b = a − 2and −α − b = −2b + a. Therefore, the transition layer has width O(ε1+α/2) and thevariations in P and T are O(ε1−α/2). Since we require these variations to be small, wemust have α < 2. We conclude that the analysis in the distinguished limit δ = O(1)is valid until δ = O(ε2).

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For the case δ = O(1), the transition layer is around drying front R(t) and weintroduce the scaling r = R(t) + εr. We also take P , T , and S as asymptotic seriesas ε→ 0+, with

S = S0(r, t) + εS1(r, t) +O(ε2),

P = P0(r, t) + εP1(r, t) +O(ε2),

T = T0(r, t) + εT1(r, t) +O(ε2).(23)

We will first show that T0(r, t) ≡ T ∗(t) and P0(r, t) ≡ P ∗(t) := PST (T ∗(t)). Todo this, we note that, in order to match our transition layer into Region i, we musthave that

(24) P0

∣∣r→−∞ → P ∗(t) and T0

∣∣r→−∞ → T ∗(t).

By substituting (23) into (2) and (3), we obtain, at O(ε−2),

(25)∂

∂r

[P0

∂P0

∂r

1 + T T0

]= 0,

∂

∂r

[∂T0

∂r(1 +A4S0)

]= 0.

We note that these equations hold in any geometry at leading order, provided that weare sufficiently far away from any geometry-induced singularities that could produceadditional derivative terms at O

(ε−2), e.g., if R(t) = O(ε) in spherical coordinates.

Integrating (25) and imposing (24) implies that

(26) T0(r, t) ≡ T ∗(t) and P0(r, t) ≡ P ∗(t).

To determine the leading-order behavior for S, we note that using (23) in (8) andexpanding yields

(27)

PST = P ∗(

1 + εβ(1 + T )

(1 + T T ∗)2T1

)+O

(ε2),

Iv = −ε(P1 −

β(1 + T )

(1 + T T ∗)2T1P

∗)S0(1− σS0)

√1 + T

1 + T T ∗+O

(ε2).

Using these along with (24), we obtain, at O(ε−1), that (1)–(3) give

−R′(t)∂S0

∂r= Ψ(P1, T1)S0(1− σS0),(28)

σP ∗R′(t)∂S0

∂r=− 1

δΨ(P1, T1)S0(1− σS0)

(1 + T T ∗

1 + T

)+ P ∗

∂2P1

∂r2,(29)

−A1(1 + T T ∗)R′(t)∂S0

∂r= A2Ψ(P1, T1)S0(1− σS0) +A3

∂

∂r

[(1 +A4S0)

∂T1

∂r

],

(30)

where

(31) Ψ(P1, T1) :=

√1 + T

1 + T T ∗

(P1 −

β(1 + T )

(1 + T T ∗)2T1P

∗).

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Finally, the matching conditions with Regions i and ii are

S0 → 1, P1 → 0, and T1 → 0 as r → −∞,(32)

S0 → 0 as r → +∞,(33)

∂P1

∂r

∣∣∣r→+∞

=∂P

∂r

∣∣∣r→R(t)

,(34)

∂T1

∂r

∣∣∣r→+∞

=∂T

∂r

∣∣∣r→R(t)

.(35)

In interpreting (32)–(35), we note that the limits where r → R(t) are matchingconditions for Regions i and ii, whereas the limits where r → ±∞ refer to matchingconditions for the transition layer.

By eliminating the terms with Ψ(P1, T1) in (28) and (29), we obtain

(36) P ∗∂2P1

∂r2=

[σP ∗ − 1

δ

(1 + T T ∗

1 + T

)]R′(t)

∂S0

∂r.

Integrating this and imposing the matching conditions (32) yield

(37)∂P1

∂r=

[1

δ

(1 + T T ∗

(1 + T )P ∗

)− σ

]R′(t)(1− S0).

Similarly, eliminating terms with Ψ(P1, T1) in (28) and (30) gives us

(38) R′(t) [A2 −A1(1 + T T ∗)]∂S0

∂r= A3

∂

∂r

[(1 +A4S0)

∂T1

∂r

].

Integrating and imposing the matching conditions (32) yields, after some rearranging,

(39)∂T1

∂r= − 1

A3R′(t) [A2 −A1(1 + T T ∗)]

(1− S0

1 +A4S0

).

Finally, by rearranging (28) to isolate S0, we obtain

(40)∂S0

∂r

S0(1− σS0)= −Ψ(P1, T1)

R′(t).

In order to write a single differential equation for S0, we differentiate (40) with respectto r, as well as substitute in (37) and (39), to give us

(41)∂2S0

∂r2−(∂S0

∂r

)21− 2σS0

S0(1− σS0)+ S0(1− σS0)(1− S0)Υ(S0) = 0,

where we define

Υ(S0) :=

√1 + T

1 + T T ∗

[1

δ

(1 + T T ∗

(1 + T )P ∗

)− σ(42)

−(

β(1 + T )P ∗

A3(1 + T T ∗)2

)(A2 −A1(1 + T T ∗)

1 +A4S0

)].

We note that, aside from the denominator 1 +A4S0, the components of the functionΥ(S0) are independent of r. By identifying (41) as a Bernoulli-like ODE, we can use

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integrating factors and the matching conditions (33) to obtain the first-order nonlinearautonomous differential equation for S0(r):

(43)∂S0

∂r= −S0(1− σS0)

√2

ˆ 1

S0

(1− χ)Υ(χ)

χ(1− σχ)dχ .

It immediately follows that S0 is monotone decreasing when S0 lies between 0 and1. Hence, we conclude that P and T do not drastically change within the transitionlayer and the O(ε) perturbations P1 and T1 can be related to S0, which is the solutionof a first-order differential equation in r.

2.3. Asymptotics of Region ii. While the leading-order dynamics of S, T,and P have been determined in the transition layer, we still do not have an explicitform for R(t) and T ∗(t). To find these, we now examine Region ii, where negligiblewater is present. From (1), we have that S = 0 at O(ε−2). However, this in turncauses a cascading effect in the asymptotic expansion of (1), and we conclude thatS = o(εn) for all natural numbers n. Motivated by this fact, we anticipate that S willbe exponentially small in Region ii. Therefore, we can solve (1) explicitly, coupledwith the condition that S = 1 at t = t∗:

(44) S =

{(1− σ) exp

[1

ε2

ˆ t

t∗(PST (T (r, s))− P (r, s))

√1 + T

1 + T T (r, s)ds

]+ σ

}−1

.

If we then neglect exponentially small terms, (2) and (3) become

∂

∂t

[P

1 + T T

]= ∇ ·

[P∇P

1 + T T

],(45)

∂T

∂t= A3∇2T.(46)

For our boundary conditions in Region ii, we have the matching conditions (33)–(35),implying

T∣∣r→R(t)

→ T ∗(t), P∣∣r→R(t)

→ P ∗(t),(47)

∂P

∂r

∣∣∣r→R(t)

=∂P1

∂r

∣∣∣r→+∞

→[

1

δ

(1 + T T ∗

(1 + T )P ∗)

)− σ

]R′(t),(48)

∂T

∂r

∣∣∣r→R(t)

=∂T1

∂r

∣∣∣r→+∞

→ − 1

A3[A2 −A1(1 + T T ∗)]R′(t).(49)

We must also give an initial condition for R(t) and T ∗(t). As the drying frontstarts from the surface of the bean and the temperature is at the switching point of(6), the initial conditions are R(t∗) = 1, T ∗(t∗) = Ta. Finally, our solutions must alsocontinue to agree with the external boundary conditions of the system, namely,

(50)∂T

∂r

∣∣∣r=1

= ν

(1 +A4

1− σ

)[1− T

∣∣r=1

]and P

∣∣r=1

= Pa.

Therefore, our leading-order problem is a coupled system of two Stefan-like problemsin a mass transfer setting, rather than the classical heat transfer setting [19]. As thiscoupled system of PDEs (45)–(50) is not explicitly solvable, we are motivated to usethe large Stefan number approximation by considering the limiting case when δ � 1.

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The large Stefan number approximation has been studied previously in the contextof heat transfer on spheres (see, e.g., [20]). For Region ii, the large Stefan limit δ � 1corresponds to the drying front moving very slowly relative to the speed of vapordiffusion.

By rescaling time with τ = δ(t− t∗), we can examine the asymptotic series

(51) T = T0(r, τ) + δT1(r, τ) +O(δ2), P = P0(r, τ) + δP1(r, τ) +O(δ2)

as δ → 0+. In consequence, our leading-order Region ii problem (45)–(50) becomes

∇2T0 = 0, T0

∣∣r→R(τ)

→ T ∗(τ),∂T0

∂r

∣∣∣r→R(τ)

→ 0,(52)

∇ ·(P0∇P0

1 + T T0

)= 0, P0

∣∣r→R(τ)

→ P ∗(τ),∂P0

∂r

∣∣∣r→R(τ)

→(

1 + T T ∗

(1 + T )P ∗

)R′(τ),

(53)

∂T0

∂r

∣∣∣r=1

= ν

(1 +A4

1− σ

)[1− T0

∣∣r=1

], P0

∣∣r=1

= Pa,(54)

R(0) = 1, T ∗(0) = Ta.(55)

Solving (52) implies that T0 ≡ T ∗(τ) and applying (54) forces T ∗(τ) ≡ 1. In con-sequence, this reduces our coupled Stefan problem to a Stefan problem for pressurealone, i.e.,

∇ · (P0∇P0) = 0, P0

∣∣r→R(τ)

→ 1, P0

∣∣r=1

= Pa,(56)

∂P0

∂r

∣∣∣r→R(τ)

→ R′(τ),(57)

R(0) = 1.(58)

We note that this solution cannot satisfy the initial condition (55) for T ∗(t). For thisto be resolved, we would have to consider the full problem in t, (45)–(50), which isnot readily solvable.

2.3.1. Determining R(t) in Cartesian coordinates with T ∗ ≡ 1. In thelimiting case where δ � 1, i.e., T ∗ ≡ 1, we can solve (56), provided that we neglectany short-time discrepancies between the initial condition T ∗(0) = Ta and T ∗ ≡ 1.Solving this PDE system (56) gives us

(59) P0(r, τ) =

√1− (1− P 2

a )

(r −R(τ)

1−R(τ)

)and our Stefan condition (57) gives us the ODE

(60)dR

dτ= − 1− P 2

a

2(1−R).

Based on the initial condition from (58), our drying front in Cartesian coordinatesbased on leading-order asymptotics, RCart(τ), is

(61) RCart(τ) = 1−√

(1− P 2a )τ .

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By returning to the original time scale of Region ii, we determine that P0 can be fullyexpressed in Cartesian coordinates as

(62) PCart0 (r, t) =

√√√√P 2a + (1− r)

√1− P 2

a

δ (t− t∗Cart).

Finally, we determine from (61) that the time to completely dry a bean based onleading-order asymptotics is

(63) tdryCart = t∗Cart +

1

δ(1− P 2a ).

Using parameter values shown in [6], as well as typical values Pa = 0.0879, δ = 0.1011,

σ = 0.0842, and t∗Cart ≈ 0.494, we compute that tdryCart ≈ 10.46, or about 2768 seconds

in dimensional units.

2.3.2. Determining R(t) in spherical coordinates with T ∗ ≡ 1. In thelimiting case where δ � 1, i.e., T ∗ ≡ 1, we have that in spherical coordinates, bysolving (56),

(64) P0(r, τ) =

√1−

(1− P 2

a

r

)(r −R(τ)

1−R(τ)

),

and our Stefan condition (57) gives us the ODE

(65)dR

dτ= − 1− P 2

a

2R(1−R).

We use our initial condition (58) to give us, in implicit form, that the inverse functionof the drying front in spherical coordinates, τSph(R), satisfies the equation

(66) τSph(R) =1−R2(3− 2R)

3(1− P 2a )

.

We can invert (66) and solve RSph(τ) in the correct domain and range:

(67) RSph(τ) =1

2

(1−

exp(

2πi3

)Ξ (3(1− P 2

a )τ)− exp

(−2πi

3

)Ξ(3(1− P 2

a )τ))

,

where

(68) Ξ(χ) =3

√2√χ(χ− 1)− 2χ+ 1

and Ξ(χ) uses the principal branch of the cube root. Now that we have determinedR(τ) in spherical coordinates, we can return to our original time scale of the problemand obtain that our leading-order asymptotic approximation for P is(69)

P Sph0 (r, t) =

√√√√√√1−(

1−P2a

r

)1− 2(1−r)

1+exp( 2πi

3 )Ξ(3δ(1−P2

a )(t−t∗Sph

))+exp(−2πi

3 )Ξ(3δ(1−P2

a )(t−t∗Sph

)).D

ownl

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d 02

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.1.8

1.26

. Red

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o SI

AM

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0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

t

Dry

ing

Fro

nt

R(t

)

Leading−OrderAsymptotics (Cartesian)Leading−OrderAsymptotics (Spherical)

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Ra

diu

s (

mm

)

Numerics

Asymptotics

(a) (b)

Fig. 2. Comparison of predictions of the drying front position R(t) for the variable temperatureregime. (a) Predictions RCart(t) from (61) and RSph(t) from (67). (b) Spherical predictions RSph(t)from (67), shown in dash-dot red, and numerical solutions of (1)–(8), shown in black. All predictionsin (b) are shown in dimensional units.

To determine the time where the bean becomes fully dry, we substitute R = 0 into(66) to obtain, in our original time scale, that

(70) tdrySph = t∗Sph +

1

3δ(1− P 2a ).

Therefore, to leading order, the time for a spherical coffee bean to dry out completelyis tdry

Sph ≈ 3.495, or about 925 seconds in dimensional units. Figure 2(a) shows acomparison between the Cartesian and spherical asymptotic approximations of R(t).

2.4. Comparison of asymptotic approximations with numerical results.We now compare these asymptotic approximations with the numerical solution of thePDE system (1)–(3), considering the drying front R(t) in particular. We solve thePDE system (1)–(3) in MATLAB using the method of lines and a second-order centralfinite difference scheme in space. We use a stiff adaptive ODE solver in time, namely,the MATLAB function ode15s, to achieve convergence. As we can see in Figure 2(b),the general shape of the dimensional drying front R(t) agrees reasonably well withthe dimensional drying front seen in the numerical solution, especially as R(t) → 0.However, we also see that the drying time in the numerical solution is larger thanthe predicted tdry

Sph from asymptotic results. This is to be expected, as the asymptotic

results used were for when the Stefan number 1δ → +∞. Therefore, for a smaller (but

still large) Stefan number, we expect the drying time to be longer. Additionally, theseapproximations for the drying front R(t) assume T ∗ ≡ 1. Because T ∗(t) is less thanunity in the numerical simulations, this will cause the drying front to be slower thanthe asymptotic approximation, which can explain why the numerical solution takeslonger to dry out the entire bean.

3. Asymptotics of the multiphase model with constant temperature. Insection 2, we have given an analysis of the leading-order equations governing Regions iand ii, and the transition layer. However, the numerical solutions indicate the thermaltime scale of the multiphase model is much smaller than the vapor diffusive time scale.In consequence, it seems reasonable to examine a reduced model where the coffee beanis at the externally imposed roasting temperature throughout. Additionally, many ofthe leading-order equations can be solved explicitly if the temperature is spatially

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uniform and this helps in interpreting the behavior of the moisture content and vaporpressure. Therefore, we are motivated from a physical and an asymptotics point ofview to consider a reduced model with T ≡ 1.

In this section, we assume that T ≡ 1 throughout the bean, which means that thetransition region is created immediately (i.e., t∗ = 0) and the system of PDEs (1)–(2)become

(71)∂S

∂t= − 1

ε2(1− P )S(1− σS),

(72)∂

∂t[(1− σS)P ] =

1

δε2(1− P )S(1− σS) +∇ · (P∇P )

with boundary conditions

(73) P∣∣r=1

= Pa,∂P

∂r

∣∣∣∣r=0

= 0,

and initial conditions

(74) S(r, 0) = 1, P (r, 0) = 1.

Formally, we will consider the asymptotics of this system in the limit as ε→ 0+. Wewill take δ = O(1) except where significant simplification is found by taking δ � 1,and all other parameters are assumed to be O(1).

3.1. Asymptotics in Region i. In Region i, we have that P = 1 to leadingorder. In fact, we note that P, S ≡ 1 is the exact solution in Region i, as these constantsolutions satisfy both PDEs, the initial conditions, and the symmetry condition in Pat r = 0. It is important to note that, since we assume that Region i is never incontact with the surface of the bean, the boundary condition at r = 1 does not apply.

3.2. Asymptotics of the transition layer. As in section 2, we introduce thescaling r = R(t) + εr to examine the behavior as S transitions from 1 to 0. Again, wecan expand P and S as asymptotic series as ε→ 0:

(75) S = S0(r, t) + εS1(r, t) +O(ε2), P = 1 + εP1(r, t) +O(ε2).

Noting that temperature is constant (implying that T1 ≡ 0 and T ∗ ≡ 1), the equation(42) for Υ(S0) reduces to Υ(S0) ≡ 1

δ − σ. From (37), this gives us

(76)∂P1

∂r=

(1

δ− σ

)R′(t)(1− S0),

and from (43), gives us the first-order nonlinear autonomous ODE

(77)∂S0

∂r= −S0(1− σS0)

√2

(1

δ− σ

)[1− σσ

log

(1− σ

1− σS0

)+ log

(1

S0

)]with matching conditions S0 → 0 as r → +∞ and S0 → 1 as r → −∞. It is importantto note a few key points about (77). First, it is not explicitly solvable. Second, dueto translational invariance, we require an additional constraint for uniqueness. Thiscan be achieved, for example, by assuming the unique inflection point of S0 occurs atr = 0. With this additional constraint, we numerically solve (77) and plot the resultsin Figure 3.

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−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

Inner Spatial Variable r

TransitionLayer

SolutionS0(r)

−3 −2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

Inner Spatial Variable r

∂S0/∂r

Fig. 3. Numerical solution of the ODE (77). The left panel shows the solution S0(r) and the

right panel shows its spatial derivative ∂S0∂r

. For uniqueness, we pick a constant of integration sothat S0(r) has an inflection point at r = 0.

3.3. Asymptotics in Region ii. In Region ii, we have, from (71), that S = 0at O(ε−2), which again causes a cascading effect in the asymptotic expansion of (71).Similarly to what was done in section 2, we find that S is exponentially small inRegion ii and is given by

(78) S =

{(1− σ) exp

[1

ε2

(t−ˆ t

0

P (r, s)ds

)]+ σ

}−1

.

Additionally, by neglecting exponentially small S, (72) becomes

(79)∂P

∂t= ∇ · (P∇P ) .

From (76), our boundary and initial conditions become(80)

P∣∣∣r=1

= Pa, P∣∣∣r→R(t)

→ 1,∂P

∂r

∣∣∣r→R(t)

→(

1

δ− σ

)R′(t), R(0) = 1.

This problem has a similarity solution in Cartesian coordinates, as will be shown insection 3.3.1, although it cannot be explicitly solved. However, we can also examinethe physically relevant large Stefan number limit by letting δ → 0+, as was donein section 2.3. By rescaling time with τ = δt and considering the asymptotic seriesP ∼ P0(r, τ) + δP1(r, τ) +O(δ2), (79)–(80) become(81)

∇ · (P0∇P0) = 0, P0

∣∣∣r=1

= Pa, P0

∣∣∣r→R(τ)

→ 1,∂P0

∂r

∣∣∣r→R(τ)

→ R′(t), R(0) = 1.

Solving (81) like in section 2, we determine that in Cartesian coordinates,

(82) P0(r, t) =

√P 2a + (1− r)

√1− P 2

a

δt, R(t) = 1−

√(1− P 2

a )δt,

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and in spherical coordinates,

(83)

P0(r, t) =

√√√√√√1−(

1− P 2a

r

)1− 2(1− r)

1 +exp( 2πi

3 )Ξ(3(1−P 2

a )δt) +Ξ(3(1−P 2

a )δt)

exp( 2πi3 )

,R(t) =

1

2

(1−

exp(

2πi3

)Ξ (3(1− P 2

a )δt)−

Ξ(3(1− P 2

a )δt)

exp(

2πi3

) ),

where Ξ(χ) = 3

√2√χ(χ− 1)− 2χ+ 1.

3.3.1. Determining R(t) in Cartesian coordinates using similarity so-lutions. One might consider using a similarity solution to solve the system (79)–(80)in Cartesian coordinates without the assumption that δ � 1. To do this, we letP = h(η), where η = 1−r√

t. Substituting this transformation into (79) gives

(84) (h(η)h′(η))′+η

2h′(η) = 0,

and (80) becomes

(85) h(0) = Pa, h(λ) = 1, h′(λ) =λ (1− δσ)

2δ.

Here, η = λ corresponds to the moving boundary R(t). Thus, our drying front basedon the Cartesian similarity solution, is given by

(86) RSS(t) = 1− λ√t.

We note that our choice of η allows us to automatically satisfy the initial conditionR(0) = 1 and we can determine from this equation when the bean will be completely

dry, i.e., when RSS(t) = 0. This gives us tdrySS = 1

λ2 . As (84) is not explicitly solvable, itis necessary to numerically solve this boundary value problem in order to determineλ. Using the shooting method, with the typical values Pa = 0.0879, δ = 0.1011,and σ = 0.0842, we find that λ ≈ 0.3152, implying that tdry

SS ≈ 10.06, or about

2664 seconds in dimensional units. With less than a 1% relative error to tdrySS , we

conclude that tdryCart ≈ 9.964, as described in (63), is a very good approximation to the

drying time computed from the similarity solution. Figure 4(a) shows a comparisonof the drying front RSS(t) with the Cartesian drying front determined previously viaasymptotic methods, namely, RCart(t) given in (61).

3.4. Comparison of asymptotic approximations with numerical results.Comparing the various asymptotic approximations with the numerical solutions of(71)–(74), we can see in Figure 4(b) that the general shape of the dimensional dryingfront R(t) agrees well with the dimensional drying front seen in the numerical solution.Because we have assumed that T ∗ ≡ 1, we no longer have differences induced byvarying the temperature that were seen in section 2. Therefore, it is expected thatthe drying front R(t) determined via asymptotics fits closer to the numerics. We see,like in section 2, that the drying time predicted by the numerical solution is largerthan tdry

Sph determined from asymptotic results in (70). However, this is to be expected;a large (but finite) Stefan number would cause the drying time to be longer than the

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0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t

Dry

ing

Fro

nt

R(t

)

Similarity Solution(Cartesian)Leading−OrderAsymptotics (Cartesian)

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Ra

diu

s (

mm

)

Numerics

Region ii PDE

Asymptotics

(a) (b)

Fig. 4. Comparison of predictions of the drying front position R(t) for the constant temperatureregime. (a) Cartesian predictions RSS(t) from (86) and RCart(t) from (82). (b) Spherical predictionsRSph(t) from (83), shown in dash-dot red, and numerical solutions of (71)–(74), shown in black,and of (79)–(80), shown in dashed blue. All predictions in (b) are shown in dimensional units.

time produced by the limit δ → 0+. This is confirmed by comparing the drying frontin the full numerical solution with the drying front obtained by solving (79)–(80)numerically. Indeed, by allowing ε → 0+, we only observe significant discrepanciesbetween these two drying fronts near the surface and the center of the bean. Thisis to be expected, as the transition layer will be on a semi-infinite region in both ofthese cases and will have different behavior due to the boundary conditions.

4. Discussion. In this paper, we have extended results of the simplified formof the multiphase model presented in [6] via asymptotic methods, in order to betterunderstand the qualitative features of the coffee bean roasting process. Motivated byprevious numerical results, we considered the limit ε→ 0+, representing the situationwhere the rate of vapor transport by Darcy flow is much smaller than the evaporationrate. The asymptotic analysis showed that the solution could be divided into twomain regions and a transition layer. The entire bean was in the first region untila time t∗, when a thin transition layer appears at the surface of the bean. Thistransition layer then propagated into the bean creating a second main region betweenit and the surface of the bean. This asymptotic limit is different from what has beenstudied previous in drying models, since the rigid cellulose structure of the solid coffeebean creates a large buildup of vapor pressure that drives the vapor to the externalenvironment. The analysis shows that a narrow drying front, represented by thetransition layer, is crucial to the drying process in this limit.

In the first region, the vapor pressure is in equilibrium with the steam tablepressure and the moisture content of the bean remains at its initial value, with heatflow governed by the heat equation. In the thin transition region, the moisture contentchanges rapidly from its initial value to a small value. Here, evaporation dominatesand the temperature and vapor pressure remain spatially uniform. Finally, in thesecond main region, there is almost no water and therefore no evaporation. Theproblem in this second region consists of diffusion equations for the heat and vaporflow with coupling through the matching conditions, similar to a Stefan problem, atthe transition layer.

Numerical simulations suggest that the externally applied roasting temperatureis attained globally fairly quickly; hence, the case where the temperature is fixedat the roasting temperature was considered. This also allowed the coupled Stefan

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problem to be reduced to a single Stefan problem, which could then be solved viasimilarity solutions or large Stefan number asymptotics. The leading-order expressionsare shown to agree well with the dynamics of the drying front found from numericalsimulations, under both spherical and planar geometries.

Despite several simplifications made in obtaining asymptotic solutions in eachof the regions of the coffee bean, a reasonable agreement between the asymptoticapproximations and the numerical solution of the multiphase model as described in[6] has been obtained. This suggests that the asymptotics found here accuratelycapture the qualitative behavior of the coffee bean roasting process, and provide anacceptable compromise between a simpler heat transfer model (such as those presentedin [5]) and more complicated multiphase models. The asymptotic results presentedin this paper can be extended in order to determine the asymptotic dynamics ofrelated heat and mass transfer models. The complete multiphase model describedin [6] incorporates variable porosity, and by using similar methods to those shownhere, one might determine the leading-order behavior of the multiphase model withvariable porosity. Additionally, the analysis on more complicated geometries, suchas nonradially symmetric perturbations of a slab or sphere, may lead to destabilizingthe drying front observed in simpler geometries. Similarly, one might use the generalasymptotic results for the multiphase model discussed here to guide the developmentof relevant solid mechanics models, which take into account the structural properties ofthe coffee bean and allow for variations in coffee quality due to structural deformationswhich may occur during heating and roasting. Asymptotic results may also guide inthe development of more complicated models involving many more chemical reactions,as well as in understanding taste and aromatic properties of the final product.

Appendix A. Derivation of the multiphase model. Here, we will derivethe multiphase model (1)–(8) discussed in this paper, starting with (42)–(49) in [6].Using the rescalings

(87) S = σS, T = T , pv = pST (1)P , t =φ

D3pST (1)t ,

and defining the parameters

ε =

√D3

√1 + T

φ, δ =

pST (1)α2

σ(1 + T ), β =

B2T

1 + T,(88)

A1 =σφ

α1C1(1− φ)T, A2 = γA1, A3 =

φ(ζ1(1− φ) + ζ3φ)

D3pST (1)α1C1(1− φ),(89)

A4 =φσ(ζ2 − ζ3)

ζ1(1− φ) + ζ3φ,(90)

the model of [6] becomes (dropping hats)

∂S

∂t= − 1

ε2Iv,(91)

∂

∂t

[(1 + T )P (1− σS)

1 + T T

]= −1

δ

∂S

∂t+∇ ·

[(1 + T )P∇P

1 + T T

],(92)

∂T

∂t+A1

∂

∂t[S(1 + T T )] = A2

∂S

∂t+A3∇ · [(1 +A4S)∇T ] .(93)

Here, the rescaled evaporation rate Iv (following a Langmuir evaporation profile [21])

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and the rescaled steam table pressure PST (T ) are given by

(94) Iv = S(1− σS)(PST − P )

√1 + T

1 + T Tand PST (T ) = exp

(β(T − 1)

1 + T T

).

We note that in [6], the initial saturation value was defined as S0. This notationmeans something different in the analysis of this paper, so we have therefore changedthe notation of the initial saturation to be denoted as σ. The boundary conditionswe impose on the PDE system (1)–(3) are the symmetry conditions at the center ofthe bean

(95) ∇T · n = 0, ∇P · n = 0 at r = 0,

as well as the heat transfer condition

(96) ∇T · n = ν

(1− σS1− σ

)(1 +A4

1 +A4S

)(1− T ) at r = 1,

where

(97) ν =Nuvζ3φ(1− σ)

(ζ1(1− φ) + ζ3φ)(1 +A4).

Previously, the model introduced in [6] imposes a Dirichlet condition in P at thesurface of the bean. We will instead impose a different boundary condition for P inorder to prevent condensation from occurring at the surface of the bean. This can beachieved by imposing that P equals the steam table pressure for temperatures belowthe evaporating temperature, i.e.,

(98) P∣∣r=1

=

{PST (T ), T < Ta,

Pa, T ≥ Ta.

Here, Pa := 1+TpST (1) and Ta := P−1

ST (Pa). We will also make the assumption that

the switching between the boundary condition for P only occurs at one critical time,namely, t∗. We define t∗ as the time when the surface temperature equals the evapo-ration temperature Ta, i.e., as the solution to the equation T (1, t∗) = Ta. Finally, weimpose the initial conditions corresponding to uniform initial moisture content, roomtemperature, and equilibrium steam table pressure, i.e.,

(99) S(r, 0) = 1, T (r, 0) = 0, P (r, 0) = PST (0).

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ppl ath vol. 78, no. 1, pp. 418{436 motivated by coffee

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